Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXsymmetric tensors, 938symmetry in equivalence relations, 1064symmetry operationson molecules, 1041order of application, 1044t-test, see Student’s t-testt substitution, 65tan −1 x, Maclaurin series for, 140tangent planes to surfaces, 346tangent vectors to space curves, 342tanh, hyperbolic tangent, see hyperbolicfunctionsTaylor series, 136–141and finite differences, 1019, 1026and Taylor’s theorem, 136–139, 853approximation errors, 139in numerical methods, 992, 1003as solution of ODE, 1023for functions of a complex variable, 853–855for functions of several real variables, 160–162remainder term, 138required properties, 136standard forms, 136tensors, see Cartesian tensors and Cartesiantensors, particular and general tensorstest statistic, 1278tetrahedral group, 1115tetrahedronmass of, 194volume of, 192thermodynamicsfirst law of, 176Maxwell’s relations, 176–178top-hat function, see rectangular distributiontorque, vector representation of, 223torsion of space curves, 342total derivative, 154total differential, 154trace of a matrix, 258and second-order tensors, 939as sum of eigenvalues, 280, 287invariance under similarity transformations,284, 1092trace formula, 287transcendental equations, 986transformation matrix, 283, 289transformationsactive and passive, 948conformal, 839–844coordinate, see coordinate transformationssimilarity, see similarity transformationstransforms, integral, see integral transforms andFourier transforms and Laplace transformstransients, and the diffusion equation, 723transitivity in equivalence relations, 1064transpose of a matrix, 250, 255product rule, 256transverse vibrationsmembrane, 677, 739, 768rod, 769string, 676trapezium rule, 1002–1004trial functionsfor eigenvalue estimation, 793for particular integrals of ODE, 494trials, 1119triangle inequality, 246, 559triangle, centroid of, 216triangular matrices, 269tridiagonal matrices, 998–1000, 1030, 1033trigonometric identities, 10–15triple integrals, see multiple integralstriple scalar product, see scalar triple producttriple vector product, see vector triple productturning point, 50uncertainty principle (Heisenberg), 435–437from commutator, 664undetermined coefficients, method of, 494undetermined multipliers, see Lagrangeundetermined multipliersuniform distribution, 1194union ∪, probability for, 1121, 1125, 1128uniqueness theoremKlein–Gordon equation, 711Laplace equation, 741Poisson equation, 705–707unit step function, see Heaviside functionunit vectors, 219unitarymatrices, 271eigenvalues and eigenvectors, 278representations, 1086transformations, 285upper triangular matrices, 269variable end-points, 782–785variable, dummy, 61variables, separation of, see separation ofvariablesvariance σ 2 , 1146from MGF, 1163from PGF, 1159of dependent RV, 1203of sample, 1224variation of parameters, 508–510variation, constrained, 785–787variational principles, physical, 787–790Fermat, 787Hamilton, 788variations, calculus of, see calculus of variationsvector operators, 347–369acting on sums and products, 354combinations of, 355–357curl, 353, 368del ∇, 348del squared ∇ 2 , 352divergence (div), 352geometrical definitions, 398–4001331